▶Japanese
Post to SNS
- Home
- > School of Science
- > Mathematics
- > Special courses on advanced topics in Mathematics C
- School of Science
-
School of Engineering
- Group 2
- Group 3
- Group 4
- Group 5
- Group 6
- Metallurgical Engineering
- Organic and Polymeric Materials
- Inorganic Materials
- Chemical Engineering Course
- Chemical Engineering Course(Chemical Engineering)
- Applied Chemistry Course(Chemical Engineering)
- Polymer Chemistry
- Mechanical Engineering and Science
- Mechanical and Intelligent Systems Engineering
- Mechano-Aerospace Engineering
- International Development Engineering
- Control and Systems Engineering
- Industrial and Systems Engineering
- Electrical and Electronic Engineering
- Computer Science
- Civil Engineering
- Architecture and Building Engineering
- Social Engineering
- Common Cource of Engineering
- School of Bioscience and Biotechnology
- Common Course of Undergraduate School
-
Graduate School of Science and Engineering
- Mathematics
- Physics(Particle- Nuclear- and Astro-Physics)
- Physics(Condensed Matter Physics)
- Chemistry
- Earth and Planetary Sciences
- Chemistry and Materials Science
- Metallurgy and Ceramics Science
- Organic and Polymeric Materials
- Applied Chemistry
- Chemical Engineering
- Common Course of Mechanical Engineering
- Mechanical Sciences and Engineering
- Mechanical and Control Engineering
- Mechanical and Aerospace Engineering
- Common Course of Electronic Engineering
- Electrical and Electronic Engineering
- Physical Electronics
- Communications and Integrated Systems
- Communications and Computer Engineering
- Civil Engineering
- Architecture and Building Engineering
- International Development Engineering
- Nuclear Engineering
- Graduate School of Bioscience and Biotechnology
-
Interdisciplinary Graduate School of Science and Engineering
- Built Environment
- Energy Sciences
- Computational Intelligence and Systems Science
- Electronic Chemistry
- Materials Science and Engineering
- Innovative and Engineered Materials
- Environmental Chemistry and Engineering
- Environmental Science and Technology
- Mechano-Micro Engineering
- Information Processing
- Electronics and Applied Physics
- Graduate School of Information Science and Engineering
- Graduate School of Decision Science and Technology
- Graduate School of Innovation Management
- Common Course of Graduate School
- Program for Leading Graduate Schools
- Academic unit or major
- Mathematics
- Instructor(s)
- Akutagawa Kazuo Takakuwa Shoichiro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive ()
- Group
- -
- Course number
- ZUA.E333
- Credits
- 2
- Academic year
- 2016
- Offered quarter
- 2Q
- Syllabus updated
- 2016/4/27
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
In this course we take up two representative topics in geometric analysis, one is harmonic map and the other is Ricci flow. Harmonic maps are the generalization of harmonic funcions, geodesics and minimal surfaces, is first developed in geometric analysis. Ricci flow is a nonlinear partial differential equation for Riemannian metrics introduced by Richard Hamilton in 1982 to find canonical metrics on manifolds. We explain basic results for both topics.
Through these two topics we understand how the method of analysis and the method of geometry are used in unified manner to solve problems in geometric analysis.
Student learning outcomes
・Be familiar with the basic techniques in partial differential equations.
・Be familiar with many basic examples of harmonic maps and solutions of Ricci flow.
・Understand basic theory of harmonic maps and associated heat flow.
・Understand basic theory of Ricci flow.
Keywords
Riemannian manifold, curvature tensor, geodesic, energy functional, harmonic map, heat equation, Einstein-Hilbert functional,
Ricci flow, Ricci soliton
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Basic concepts of Riemannian geometry | Details will be provided during each class session |
| Class 2 | Length of curves and geodesics | Details will be provided during each class session |
| Class 3 | Existence theorem of geodesics | Details will be provided during each class session |
| Class 4 | The 1st variation formula of energy of curves | Details will be provided during each class session |
| Class 5 | The 2nd variation formula of energy of curves | Details will be provided during each class session |
| Class 6 | Applications to geometry of geodesics | Details will be provided during each class session |
| Class 7 | Definitions and examples of harmonic maps | Details will be provided during each class session |
| Class 8 | Existence theorem of harmonic maps | Details will be provided during each class session |
| Class 9 | Applications to geometry of harmonic maps | Details will be provided during each class session |
| Class 10 | Overview of Hamilton's Ricci flow ( definition, background and history ) | Details will be provided during each class session |
| Class 11 | Various solutions of Ricci flow | Details will be provided during each class session |
| Class 12 | Existence and uniqueness of short time solutions of Ricci flow | Details will be provided during each class session |
| Class 13 | Evolution of curvature | Details will be provided during each class session |
| Class 14 | Convergence of Ricci lfow | Details will be provided during each class session |
| Class 15 | Applications to geometry of Ricci flow | Details will be provided during each class session |
Textbook(s)
none required
Reference books, course materials, etc.
・B．Andrews-C．Hopper, The Ricci flow in Riemannian Geometry, Lect. Notes in Math. Vol. 2011, Springer, 2011
Assessment criteria and methods
reports 100%
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
Prerequisites (i.e., required knowledge, skills, courses, etc.)
manifold theory, calculus of several variables, functional analysis
